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Other The math required to mechanics course

  1. Jun 22, 2016 #1
    Hi,
    This October I'll open my physics undergraduate studies :D ♥♥♥
    And as you all know, first things first - mechanics.
    I realize that it includes dealing with simple differential equations and that the accompanying mathematical explanations are not totally satisfactory. Therefore, I intend to purchase a little pre-self-experience. Nothing crazy, a little practice.
    Can you give me a hand by pointing the relevant families of DE I'll ran into?
    Thank you all.
     
  2. jcsd
  3. Jun 22, 2016 #2

    vanhees71

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    The usual examples are

    -free particle
    -particle in the gravitational field of the earth (approximated to be ##\vec{g}=\text{const}##)
    -harmonic oscillator (damped and undamped, driven and undriven)
    -energy, momentum, angular momentum, center-mass motion (general conservation laws)
    -Kepler problem (two celestial bodies moving around their common center of mass due to the gravitational interaction, i.e.)
     
  4. Jun 22, 2016 #3
    Thanks!
    Let me clarify myself: I'm looking after the families of DE, so I will be able to look for it on a math textbook, find it and practice it regardless it's possible physics uses..
    Should I be interested in first or second order equations? etc.
     
  5. Jun 22, 2016 #4

    vanhees71

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    A nice book for this purpose is

    R. Bronson, Differential Equations, Schaum’s Easy Outlines, McGraw-Hill, New York, Chicago,
    San Francisco (2003).

    You should look at 1st-order equations, because usually you try to use the conservation laws as much as you can to reduce a mechanics problem to a set of 1st order ODEs.

    Of course, the full glory of theoretical mechanics will become clear only in the next semester, when analytical mechanics is treated. There you'll learn to make use of systematic methods using Lie-group theory (Hamilton canonical equations, Poisson brackets, Noether's theorem) and all that :-).
     
  6. Jun 22, 2016 #5
    Thank you for a lovely answer.
    So, I have no interest of any second dgree or order. Is that correct?

    About your remark on analytical mechanics, well, I'm Looking forward to it, be sure ;)
     
  7. Jun 22, 2016 #6

    jedishrfu

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  8. Jun 22, 2016 #7

    jtbell

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    Maybe things are different in your country, but in the US, first-year university physics courses generally do not require solving differential equations. They require only basic differential and integral calculus. Have you seen a syllabus for your course that indicates otherwise?
     
  9. Jun 22, 2016 #8
    I know for certain that some universities here (Israel) uses DE at least for harmonic oscillator..
    Maybe my best move will be to contact a teacher on that specific university and ask him...
     
  10. Jun 22, 2016 #9
    For these harmonic oscillators the differential equation needed is really simplistic. I believe they are of the basic homogenous equations. I can be wrong, it has been a long time since I did this. But it did not require a full differential course, just 15 min to learn how to do it.
     
  11. Jun 22, 2016 #10

    jtbell

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    At the first-year level, many intro courses (at least in the US) use e.g. the harmonic oscillator to introduce the concept of a differential equation, and "solve" it in a hand-waving way without using the rigorous techniques of a DE course. We write something like $$F = ma \\ -kx = ma \\ -kx = m \frac {d^2 x} {dt^2} \\ \frac {d^2 x} {dt^2} = -\frac k m x$$ and then we ask, "OK, what function x(t) has a second derivative that's proportional but with a minus sign?" Someone answers "sine", someone else answers "cosine". We substitute those and see that they work. Then we see what ways we can introduce arbitrary constants.

    In a later course, students learn the rigorous way to solve this DE.

    Things may be different elsewhere, so ask your instructor, if you can.
     
    Last edited: Jun 26, 2016
  12. Jun 25, 2016 #11

    chiro

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    Hey TamirN.

    Does your college/university have a co-requisite or pre-requisite for mathematics courses involving differential equations?

    A proper course on differential equations will deal with analytic and numerical techniques and they will go quite into depth. They usually build off the calculus courses which contain a couple of simple DE's to solve (relative to the more involved course).

    Universities often go out of their way to provide everything that is necessary in a course and the course structure (in terms of pre-requisites and co-requisites) should reflect that.
     
  13. Jun 25, 2016 #12

    micromass

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    I'm not sure why you are so worried about differential equations. Differential equations are quite literally the easiest part of the mechanics course. There is very little to them (in a mechanics course) aside from knowing the correct trick to solve them, which is usually given in detail in the course itself. I can think of much better ways to prepare than studying DE's.
     
  14. Jun 26, 2016 #13
    it depends on grade of the university, for weak universities that is true
     
  15. Jun 26, 2016 #14

    micromass

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    It's true for basically any university. Even Kleppner uses little differential equations in his book and tells you how to solve them. You definitely don't need them beforehand.
     
  16. Jun 26, 2016 #15
    Here is an example of very simple problem that nevertheless requires the understanding what is qualitative analysis of ODE
    Let a a small stone of mass ##m## is thrown in the standard gravity field ##\boldsymbol g##; the air resistance force is proportional to the square of the velocity of the stone i.e. ##\boldsymbol F=-k|\boldsymbol v|\boldsymbol v,\quad k=const>0##. Show that the velocity has a limit as ##t\to\infty## and the trajectory of the stone tends to a vertical line asymptotically
     
    Last edited: Jun 26, 2016
  17. Jun 26, 2016 #16

    ZapperZ

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    I really dislike these questions on "what math do I need to so such-and-such". It is as if one can easily pick and choose the mathematics that one will need If you are a physics or engineering major, you need a bunch of math to be able to survive the entire program.

    I strongly suggest:

    1. You study Mathematics needed for your physics major, rather than specific areas.

    2. You do a search on Mary Boas, and her text "Mathematical Methods in the Physical Sciences". There have been numerous threads created on this topic, on that text in particular. The chapter on Variational Mathods alone, which is relevant to Classical and Quantum mechanics, is worth the price of that text.

    Zz.
     
    Last edited: Jun 26, 2016
  18. Jun 26, 2016 #17

    Vanadium 50

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    I do as well. In particular the subtext: "I want to learn the exact minimum for X". That's not a good attitude, and besides, for every mathematical tool used, there was a first person to use it for that particular problem.
     
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